Optimal. Leaf size=326 \[ -\frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+2 e^2}}\right )}{\sqrt{g}}-\frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{\sqrt{d^2 g+2 e^2}+d \sqrt{g}}\right )}{\sqrt{g}}+\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+2 e^2}}+1\right )}{\sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{\sqrt{d^2 g+2 e^2}+d \sqrt{g}}+1\right )}{\sqrt{g}}+\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )^2}{2 \sqrt{g}} \]
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Rubi [A] time = 0.420563, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {215, 2404, 12, 5799, 5561, 2190, 2279, 2391} \[ -\frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+2 e^2}}\right )}{\sqrt{g}}-\frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{\sqrt{d^2 g+2 e^2}+d \sqrt{g}}\right )}{\sqrt{g}}+\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+2 e^2}}+1\right )}{\sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{\sqrt{d^2 g+2 e^2}+d \sqrt{g}}+1\right )}{\sqrt{g}}+\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )^2}{2 \sqrt{g}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 2404
Rule 12
Rule 5799
Rule 5561
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{2+g x^2}} \, dx &=\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}-(b e n) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}{\sqrt{g} (d+e x)} \, dx\\ &=\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}-\frac{(b e n) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}{d+e x} \, dx}{\sqrt{g}}\\ &=\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{x \cosh (x)}{\frac{d \sqrt{g}}{\sqrt{2}}+e \sinh (x)} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )\right )}{\sqrt{g}}\\ &=\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )^2}{2 \sqrt{g}}+\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{e^x x}{e e^x+\frac{d \sqrt{g}}{\sqrt{2}}-\frac{\sqrt{2 e^2+d^2 g}}{\sqrt{2}}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )\right )}{\sqrt{g}}-\frac{(b e n) \operatorname{Subst}\left (\int \frac{e^x x}{e e^x+\frac{d \sqrt{g}}{\sqrt{2}}+\frac{\sqrt{2 e^2+d^2 g}}{\sqrt{2}}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )\right )}{\sqrt{g}}\\ &=\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )^2}{2 \sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (1+\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{2 e^2+d^2 g}}\right )}{\sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (1+\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}+\sqrt{2 e^2+d^2 g}}\right )}{\sqrt{g}}+\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}+\frac{(b n) \operatorname{Subst}\left (\int \log \left (1+\frac{e e^x}{\frac{d \sqrt{g}}{\sqrt{2}}-\frac{\sqrt{2 e^2+d^2 g}}{\sqrt{2}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )\right )}{\sqrt{g}}+\frac{(b n) \operatorname{Subst}\left (\int \log \left (1+\frac{e e^x}{\frac{d \sqrt{g}}{\sqrt{2}}+\frac{\sqrt{2 e^2+d^2 g}}{\sqrt{2}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )\right )}{\sqrt{g}}\\ &=\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )^2}{2 \sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (1+\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{2 e^2+d^2 g}}\right )}{\sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (1+\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}+\sqrt{2 e^2+d^2 g}}\right )}{\sqrt{g}}+\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{\frac{d \sqrt{g}}{\sqrt{2}}-\frac{\sqrt{2 e^2+d^2 g}}{\sqrt{2}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}\right )}{\sqrt{g}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{e x}{\frac{d \sqrt{g}}{\sqrt{2}}+\frac{\sqrt{2 e^2+d^2 g}}{\sqrt{2}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}\right )}{\sqrt{g}}\\ &=\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )^2}{2 \sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (1+\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{2 e^2+d^2 g}}\right )}{\sqrt{g}}-\frac{b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \log \left (1+\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}+\sqrt{2 e^2+d^2 g}}\right )}{\sqrt{g}}+\frac{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{g}}-\frac{b n \text{Li}_2\left (-\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{2 e^2+d^2 g}}\right )}{\sqrt{g}}-\frac{b n \text{Li}_2\left (-\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}+\sqrt{2 e^2+d^2 g}}\right )}{\sqrt{g}}\\ \end{align*}
Mathematica [A] time = 0.231515, size = 275, normalized size = 0.84 \[ \frac{-2 b n \text{PolyLog}\left (2,\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{\sqrt{d^2 g+2 e^2}-d \sqrt{g}}\right )-2 b n \text{PolyLog}\left (2,-\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{\sqrt{d^2 g+2 e^2}+d \sqrt{g}}\right )+\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right ) \left (2 a+2 b \log \left (c (d+e x)^n\right )-2 b n \log \left (\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{d \sqrt{g}-\sqrt{d^2 g+2 e^2}}+1\right )-2 b n \log \left (\frac{\sqrt{2} e e^{\sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )}}{\sqrt{d^2 g+2 e^2}+d \sqrt{g}}+1\right )+b n \sinh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{2}}\right )\right )}{2 \sqrt{g}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.852, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) ){\frac{1}{\sqrt{g{x}^{2}+2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{g x^{2} + 2} b \log \left ({\left (e x + d\right )}^{n} c\right ) + \sqrt{g x^{2} + 2} a}{g x^{2} + 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \log{\left (c \left (d + e x\right )^{n} \right )}}{\sqrt{g x^{2} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt{g x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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